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Integration: Power Rule, Definite Integrals and the Trapezium Rule

Integration reverses differentiation: it finds areas under curves and recovers a function from its gradient. For the TMUA you integrate powers of x by hand, evaluate definite integrals exactly, combine ranges, and judge whether the trapezium rule over- or under-estimates.

Part of the TMUA syllabus — revision for the Test of Mathematics for University Admission (TMUA), the UAT-UK maths test used by Cambridge, Oxford, Imperial, UCL, LSE, Warwick and Durham. No calculator; multiple choice.

Key points

  • Power rule: add one to the power and divide by the new power (integral xn dx = xn+1/(n+1) + C), valid for every rational n except n = -1, where the rule breaks down.
  • Indefinite integrals always carry an arbitrary +C. To solve dy/dx = f(x) you integrate to get y = integral f(x) dx + C, then substitute a known point on the curve to pin down C.
  • Simplify BEFORE integrating: expand brackets, split fractions term by term, and rewrite roots as fractional powers (√(x) = x1/2, 1/x2 = x-2) so each term is a single power of x.
  • A definite integral equals the SIGNED area between the curve and the x-axis: regions below the axis contribute negatively, so for a total physical area you split at the roots and add the magnitudes.
  • Fundamental Theorem of Calculus: differentiation and integration are inverse operations, so integral a to b f(x) dx = F(b) - F(a) for any antiderivative F.
  • Combining ranges: integral a to b + integral b to c = integral a to c (contiguous limits join up), reversing the limits flips the sign, and integral a to a = 0.
  • Trapezium rule with n strips of width h = (b-a)/n uses n+1 ordinates: A ~= (h/2)[first + last + 2*(sum of the middle ordinates)].
  • Judging the estimate: where the curve is concave up (bends upward, second derivative > 0) the straight tops lie above it so the trapezium rule OVERESTIMATES; concave down gives an UNDERESTIMATE.

Diagram

xy01224y = x^2 + 1
A definite integral is the area between the curve and the x-axis: here the shaded region under y = x2 + 1 from x = 0 to x = 2.

Formulae

integral xn dx = xn+1/(n+1) + C

Integrating any single power of x; requires n rational and n ≠ -1. Apply once the term is written as x to a power.

integral a to b f(x) dx = F(b) - F(a)

Evaluating a definite integral or an area once you have an antiderivative F; substitute the top limit minus the bottom limit.

integral a to b f + integral b to c f = integral a to c f

Joining two integrals that share the middle limit b into one over the whole contiguous range a to c (or splitting one the other way).

integral a to b f = - integral b to a f

Swapping the limits of a definite integral; the value changes sign. Useful to reuse a result you already computed.

A ~= (h/2)[y0 + yn + 2(y1 + ⋯ + yn-1)], h = (b-a)/n

Estimating the area under a curve from n+1 equally spaced ordinates when you cannot (or need not) integrate exactly.

dy/dx = f(x) ⇒ y = integral f(x) dx + C

Solving a first-order differential equation of this form; find C by substituting a given point on the curve.

Definitions

Indefinite integral
The general antiderivative of a function, written with an arbitrary constant +C because differentiating any constant gives 0, so infinitely many functions share the same derivative.
Definite integral
A single number, integral a to b f(x) dx = F(b) - F(a), equal to the signed area between the curve y = f(x) and the x-axis from x = a to x = b.
Fundamental Theorem of Calculus
The statement that integration and differentiation are inverse processes, so a definite integral is evaluated by taking any antiderivative at the two limits and subtracting.
Trapezium rule
A method that estimates a definite integral by cutting [a, b] into equal-width strips and replacing the curve on each strip by a straight line, i.e. a trapezium.

Worked examples

1

Evaluate the definite integral of (x + 1)/√(x) with respect to x from x = 1 to x = 4, giving an exact answer.

  1. 1

    Split into powers of x:

    f = x1/2 + x-1/2
  2. 2

    Integrate term by term:

    F = (2/3)x3/2 + 2x1/2
  3. 3

    Evaluate at the top limit:

    F(4) = 16/3 + 4 = 28/3
  4. 4

    Evaluate at the bottom limit:

    F(1) = 2/3 + 2 = 8/3
  5. 5

    Subtract (FTC):

    answer = 28/3 - 8/3 = 20/3

Answer: 20/3

2

Use the trapezium rule with 4 strips to estimate the area under y = x2 + 1 from x = 0 to x = 4, and state whether it is an overestimate or an underestimate.

  1. 1

    List the 5 ordinates (h = 1):

    y = 1, 2, 5, 10, 17
  2. 2

    Apply the rule:

    A = (1/2)[1 + 17 + 2(2 + 5 + 10)]
  3. 3

    Simplify:

    A = (1/2)(52) = 26
  4. 4

    Check concavity:

    y'' = 2 > 0, so the curve is concave up
  5. 5

    Concave up means each straight top sits above the curve, so the estimate is too large.

Answer: Estimate A = 26, an overestimate (true area = 76/3 ~= 25.3).

Common mistakes

  • ×Forgetting the +C on an indefinite integral. Without it you cannot solve dy/dx = f(x), because there is no constant to fix using the given point.
  • ×Using the power rule at n = -1: you cannot write integral x-1 dx = x0/0, since dividing by zero is undefined; the rule only applies for n ≠ -1.
  • ×Integrating a product or quotient term by term without simplifying first. The integral of a product is NOT the product of the integrals, so expand or split into separate powers before integrating.
  • ×Confusing a definite integral with total area: when part of the curve is below the x-axis its contribution is negative, so integrating straight through undercounts the area. Find the roots and integrate each piece separately.
  • ×Mixing up the trapezium over/underestimate rule. Concave up (a smiley curve) always OVERestimates; concave down UNDERestimates.

No-calculator tips

  • ✓Integrate in your head with 'add one to the power, divide by the new power', then sanity-check by differentiating your answer back to the original.
  • ✓Keep every answer exact as a fraction or surd (e.g. 20/3, not 6.67); TMUA options are almost always exact, so a decimal wastes time and invites rounding slips.
  • ✓For a 'total area' question, sketch the curve and find its roots first to see whether any part dips below the axis before you integrate.
  • ✓Read off concavity from a quick sketch (or the sign of the second derivative) to call the trapezium over/underestimate instantly, without ever computing the true area.
  • ✓Reuse work: reverse a sign or join contiguous limits to build the integral you need from one you have already found, rather than re-integrating from scratch.

Test yourself

Original practice questions, no calculator. Work each out before revealing the answer.

Q1.The region R is enclosed by the curve y = x2 - 2x and the x-axis, between x = 0 and x = 2. What is the area of R?

  • A. 8/3
  • B. -4/3
  • C. 4/3
  • D. 2/3
Show answer

Answer: C — 4/3

On 0 < x < 2 the curve lies below the x-axis, so the definite integral is [x3/3 - x2] from 0 to 2 = -4/3; the area is its modulus, 4/3. Choosing -4/3 is the classic trap of treating a signed integral as an area.

Q2.The function g is defined by g(x) = integral from 2 to x of (t2 + 1) dt. What is the value of g'(3)?

  • A. 6
  • B. 9
  • C. 22/3
  • D. 10
  • E. 12
Show answer

Answer: D — 10

By the Fundamental Theorem of Calculus g'(x) = x2 + 1 (the integrand), so g'(3) = 9 + 1 = 10. The 6 comes from wrongly differentiating the integrand (2x at x=3), and 22/3 from computing g(3) itself instead of g'(3).

Q3.The trapezium rule with two strips of equal width is used to estimate integral from 0 to 2 of x2 dx (ordinates at x = 0, 1, 2). Which statement is correct?

  • A. The estimate is 3, and it is an over-estimate.
  • B. The estimate is 3, and it is an under-estimate.
  • C. The estimate is 8/3, and it is exact.
  • D. The estimate is 6, and it is an over-estimate.
  • E. The estimate is 4, and it is an over-estimate.
Show answer

Answer: A — The estimate is 3, and it is an over-estimate.

Estimate = (h/2)[y0 + 2y1 + y2] = (1/2)(0 + 2*1 + 4) = 3, versus the true value 8/3 approx 2.67; since y = x2 is convex (curves upward) the trapezia lie above it, so it over-estimates. The 6 drops the h/2 factor and the 4 uses a single strip, ignoring the middle ordinate.

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